Two ladders, one 4m long the other 5m long, are placed against opposite walls of an alleyway and in opposite directions.

If they cross at a height of 2m how far apart are the walls?

]]>If they cross at a height of 2m how far apart are the walls?

A cube is inscribed in a sphere of diameter 1 metre.

What is the surface area of the cube?

In the grid below, each of the blank squares and the square marked X are to be filled by the mean of the two numbers in its adjacent squares.

What number should go in the square marked X?

]]>What number should go in the square marked X?

Photograph: Whit Preston/Getty Images

Shown above are four men buried up to their necks in the ground.

They cannot move, so they can only look forward.

Between A and B is a brick wall which cannot be seen through.

They all know that between them they are wearing four hats--two black and two white--but they do not know what colour they are wearing.

Each of them know where the other three men are buried.

In order to avoid being shot, one of them must call out to the executioner the colour of their hat.

If they get it wrong, everyone will be shot.

They are not allowed to talk to each other and have 10 minutes to fathom it out.

After one minute, one of them calls out.

**Question:** Which one of them calls out? Why is he 100% certain of the colour of his hat?

]]>They cannot move, so they can only look forward.

Between A and B is a brick wall which cannot be seen through.

They all know that between them they are wearing four hats--two black and two white--but they do not know what colour they are wearing.

Each of them know where the other three men are buried.

In order to avoid being shot, one of them must call out to the executioner the colour of their hat.

If they get it wrong, everyone will be shot.

They are not allowed to talk to each other and have 10 minutes to fathom it out.

After one minute, one of them calls out.

1. Over the course of the whole 12 days, how many presents were sent in total?2. How many would have been if the presents continued to be given for 100 days?Justify your answers! | On the first day of Christmas, my true love sent to me A partridge in a pear tree. On the second day of Christmas, my true love sent to me Two turtle doves, And a partridge in a pear tree. On the third day of Christmas, my true love sent to me Three French hens, Two turtle doves, And a partridge in a pear tree. etc. |

The factorial on n, written n!, is defined by n! = 1 x 2 x 3 x ... x (n-2) x (n-1) x n.

For how many positive integer values of k less than 50 is it possible to find a value of n such that n! ends in exactly k zeros?

Eight identical regular octagons are placed edge to edge in a ring in such a way that a symmetrical star is formed by the interior edges.

If each octagon has side length 1, what is the area of the star?

]]>If each octagon has side length 1, what is the area of the star?

A number is chosen from the grid above and you are asked to guess what the number is?

If your guess is too high then then your number, and all higher numbers in the grid, are removed.

If your guess is too low then then your number, and all lower numbers in the grid, are removed.

1. How many attempts do you need to guarantee a correct result?

2. What is the expected number of attempts required for a correct answer?

]]>If your guess is too high then then your number, and all higher numbers in the grid, are removed.

If your guess is too low then then your number, and all lower numbers in the grid, are removed.

1. How many attempts do you need to guarantee a correct result?

2. What is the expected number of attempts required for a correct answer?